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Bending Theories.                           437

In all cases of compound stress there appear to be points of
difference between practice and the theories adopted, Thus a com-
plete theory of bending ought to simultaneously consider tension,
compression, vertical shear, and hori2ontal shear, while the bending
theory treated on p. 428 deals only with the moment of the direct
stresses \ hence we should not be surprised to find the differ-
ences referred to. Some writers show that longitudinal shear is
exceedingly small as compared with the effect of direct stress

(as also is vertical shear, in beams where the ratio of length to
breadth is considerable), Others, again, assert they have found
theory and practice agree perfectly within the elastic limit of truly
elastic material. Without doubt the greatest portion of the dis-
crepancy is due to the attempt to use the beam formula during
the plastic stage, for which it was never constructed. Neither
-can it be correct to proportion beams by using a factor on the
modulus of rupture, the method largely adopted up to say 1875
or 1880, Probably a somewhat higher value of/ may be used
than/t (but not so high a. one as^), upon which to use the safety
factor im the case of solid beams ; while thin built-up sections
irtay be proportioned by putting the safe/t in the usual formula.
The greatest dilference between^ and fQ occurs, as we should
expect, with materials tliat have no |rue elastic stage, or, what is
the same thing, with such as have a constantly varying value for
E. Among these may be mentioned cast iron, india-rubber, and
some woods (see a ho p. 453).

We have now completed our investigations of moment of
resistance, and shall proceed to consider the left side of the
lending equation.

Bending Moment and Vertical Shear.In long beams
ihe shear is small in comparison with bending stress, and is fully
met by the surplus section. For the distribution of shear stress